Ergodic equivalence relations, cohomology, and von Neumann algebras. I

Abstract

Let ( X , B ) (X,\mathcal {B}) be a standard Borel space, R ⊂ X × X R \subset X \times X an equivalence relation ∈ B × B \in \mathcal {B} \times \mathcal {B} . Assume each equivalence class is countable. Theorem 1: ∃ \exists a countable group G of Borel isomorphisms of ( X , B ) (X,\mathcal {B}) so that R = { ( x , g x ) : g ∈ G } R = \{ (x,gx):g \in G\} . G is far from unique. However, notions like invariance and quasi-invariance and R-N derivatives of measures depend only on R, not the choice of G. We develop some of the ideas of Dye [1], [2] and Krieger [1]-[5] in a fashion explicitly avoiding any choice of G; we also show the connection with virtual groups. A notion of “module over R” is defined, and we axiomatize and develop a cohomology theory for R with coefficients in such a module. Surprising application (contained in Theorem 7): let α , β \alpha ,\beta be rationally independent irrationals on the circle T \mathbb {T} , and f Borel: T → T \mathbb {T} \to \mathbb {T} . Then ∃ \exists Borel g , h : T → T g,h:\mathbb {T} \to \mathbb {T} with f ( x ) = ( g ( a x ) / g ( x ) ) ( h ( β x ) / h ( x ) ) f(x) = (g(ax)/g(x))(h(\beta x)/h(x)) a.e. The notion of “skew product action” is generalized to our context, and provides a setting for a generalization of the Krieger invariant for the R-N derivative of an ergodic transformation: we define, for a cocycle c on R with values in the group A, a subgroup of A depending only on the cohomology class of c, and in Theorem 8 identify this with another subgroup, the “normalized proper range” of c, defined in terms of the skew action. See also Schmidt [1].